Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock'

If you are trying to connect mysql using password or without password
like mysql -uUSERNAME or mysql -uUSERNAME -pPASSWORD
and getting error:-  
Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock' (2)

find first socket location
sudo find / -type s
my be location would be /tmp/mysql.sock

just go in the mysql file my.cnf
sudo nano  /etc/mysql/my.cnf

and do changes in scocket
port            = 3306
socket          = /tmp/mysql.sock

[mysqld_safe]
socket          = /tmp/mysql.sock
nice            = 0

Restart mysql, sudo /etc/init.d/mysql restart

And try login again mysql -uUSERNAME
Hope it will work.

Command to know all the mysql variable
sudo mysqladmin variables

To know mysql socket path
mysqld --verbose --help | grep ^socket